The Gamma-exponential distribution is a continuous probability distribution that combines the exponential and gamma distributions. It is also known as the Generalized Exponential Distribution or GE distribution. The GE distribution can model a wide range of hazard rate functions, including unimodal and bathtub-shaped hazard rates, making it a versatile tool in survival analysis. The GE distribution has been used in reliability analysis, finance, economics, engineering and other fields.
Gamma-exponential distribution properties
The probability density function (PDF) of the GE distribution is given by :
Where t > 0 and
- r = shape parameter, 0 ≤ r ≤ 1
- ν = concentration parameter, which measures the extent to which probability is concentrated near the mean; ν ≠ 0
- δ = scale parameter > 0
- E = (r, ν, s, t, δ)
- G = the gamma-exponential function given by 
Other parameterizations appear in the literature. For example, Tahir et al.  describe the exponential-gamma distribution as having the PDF
Gamma-exponential vs. gamma or exponential
The Gamma Exponential distribution can be considered as a more flexible distribution that combines the goodness of both the exponential and gamma distributions. When data sets have multiple modes or a bathtub-shaped hazard rate, the GE distribution can provide a better fit than either the exponential or the gamma distribution, which assume a monotonic hazard rate. In these cases, the shape parameter p of the GE distribution can capture different shapes of the hazard rate, allowing for more accurate modeling of complex data sets. Therefore, the GE distribution could be preferable over the exponential and gamma distributions when the hazard rate is non-monotonic. However, the choice of the appropriate distribution depends on the nature of the data and the research question at hand, and requires careful statistical analysis.
 Kudryavtsev, A. & Shestakov, O. Asymptotically Normal Estimators for the Parameters of the Gamma-Exponential Distribution. Mathematics 2021, 9(3), 273; https://doi.org/10.3390/math9030273. Image used via Creative Commons Attribution 4.0 International.
 Kudryavtsev, A.A.; Titova, A.I. Gamma-exponential function in Bayesian queueing models. Inform. Appl. 2017, 11, 104–108.
 Tahir, M., Cordeiro, G., Alzaatreh, A., Mansoor, M. and Zubair,M. The Logistic-X Family of Distributions and Its Applications. Communication in Statistics- Theory and Methods. 45 (24), pp 1-26. Oct. 2014